# A disease has hit a city. The percentage of the population infected t days after the disease...

## Question:

A disease has hit a city. The percentage of the population infected t days after the disease arrives is approximated by {eq}p(t) = 11te^{-\frac{t}{8}}{/eq} for {eq}0 \leq t \leq 48{/eq}.

a) After how many days is the percentage of infected people reaches a maximum?

b) What is the maximum percent of the population infected?

## Derivative:

To find the maximum population we will first find the time when the population will be maximum and it can be found by first finding the derivative of the function and then putting it equal to 0.

To find the maximum population we will find the derivative:

{eq}p(t)=11te^{\frac{-t}{8}} {/eq}

Now differentiating it we get:

{eq}p'(t)=11e^{\frac{-t}{8}}-\frac{11t}{8}e^{\frac{-t}{8}}=0\\ t=8 {/eq}

b) Now we will find the population when t=8:

{eq}P=11(8)e^{-1}\\ =32.37 {/eq}